25 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Cardinality | Obsidian::STEM | set::cardinality |
|
Equinumerosity
We say set A is equinumerous to set B, written (A \approx B) if and only if there exists a set/functions#Injections function from A set/functions#Surjections B.
%%ANKI
Basic
Suppose A is equinumerous to B. How does Enderton denote this?
Back: A \approx B
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What does it mean for A to be equinumerous to B?
Back: There exists a bijection between A and B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose A \approx B. Then what must exist?
Back: A bijection between A and B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Suppose there exists a one-to-one function F from A into B. When does this imply A \approx B?
Back: When F is also onto B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose there exists a function F from A onto B. When does this imply A \approx B?
Back: When F is also one-to-one.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose there exists a one-to-one function F from A onto B. When does this imply A \approx B?
Back: Always, by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Power Sets
No set is equinumerous to its set/index#Power Set Axiom. This is typically shown using a diagonalization argument.
%%ANKI Basic What basic set operation produces a new set the original isn't equinumerous to? Back: The power set operation. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic What kind of argument is typically made to prove no set is equinumerous to its power set? Back: A diagonalization argument. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic Who is attributed the discovery of the diagonalization argument? Back: Georg Cantor. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let g \colon A \rightarrow \mathscr{P}A. Using a diagonalization argument, what set is not in \mathop{\text{ran}}(g)?
Back: \{ x \in A \mid x \not\in g(x) \}
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Let g \colon A \rightarrow \mathscr{P}A. Why isn't B = \{x \in A \mid x \not\in g(x) \} in \mathop{\text{ran}}(g)?
Back: For all x \in A, x \in B \Leftrightarrow x \not\in g(x).
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Equivalence Concept
For any sets A, B, and C:
A \approx A;- if
A \approx B, thenB \approx A; - if
A \approx BandB \approx C, thenA \approx C.
Notice though that \{ \langle A, B \rangle \mid A \approx B \} is not an equivalence relation since the equivalence concept of equinumerosity concerns all sets.
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Basic
Concisely state the equivalence concept of equinumerosity in Zermelo-Fraenkel set theory.
Back: For all sets A, B, and C:
A \approx A;A \approx B \Rightarrow B \approx A;A \approx B \land B \approx C \Rightarrow A \approx CReference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Concisely state the equivalence concept of equinumerosity in von Neumann-Bernays set theory.
Back: Class \{ \langle A, B \rangle \mid A \approx B \} is reflexive on the class of all sets, symmetric, and transitive.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What is the reflexive property of equinumerosity in FOL?
Back: \forall A, A \approx A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
What is the symmetric property of equinumerosity in FOL?
Back: \forall A, B, A \approx B \Rightarrow B \approx A
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the transitive property of equinumerosity in FOL?
Back: \forall A, B, C, A \approx B \land B \approx C \Rightarrow A \approx C
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
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Basic
Is \{ \langle A, B \rangle \mid A \approx B \} a set?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't \{ \langle A, B \rangle \mid A \approx B \} a set?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Is \{ \langle A, B \rangle \mid A \approx B \} an equivalence relation?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Why isn't \{ \langle A, B \rangle \mid A \approx B \} an equivalence relation?
Back: Because then the field of this "relation" would be a set.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Finiteness
A set is finite if and only if it is equinumerous to a natural-numbers. Otherwise it is infinite.
%%ANKI Basic How does Enderton define a finite set? Back: As a set equinumerous to some natural number. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI Basic How does Enderton define an infinite set? Back: As a set not equinumerous to any natural number. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
Is n \in \omega a finite set?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Why isn't n \in \omega a finite set?
Back: N/A. It is.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Is \omega a finite set?
Back: No.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Why isn't \omega a finite set?
Back: There is no natural number equinumerous to \omega.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Pigeonhole Principle
No natural number is equinumerous to a proper subset of itself. More generally, no finite set is equinumerous to a proper subset of itself.
Likewise, any set equinumerous to a proper subset of itself must be infinite.
%%ANKI
Basic
How does Enderton state the pigeonhole principle for \omega?
Back: No natural number is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic How does Enderton state the pigeonhole principle for finite sets? Back: No finite set is equinumerous to a proper subset of itself. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m \in n \in \omega. What principle precludes m \approx n?
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S be a set and n \in \omega such that S \approx n. For m \in \omega, when might S \approx m?
Back: Only if m = n.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the generalization of the pigeonhole principle for \omega?
Back: The pigeonhole principle for finite sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
What is the specialization of the pigeonhole principle for finite sets?
Back: The pigeonhole principle for \omega.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What name is given to the following theorem? \text{No finite set is equinumerous to a proper subset of itself.}
Back: The pigeonhole principle.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S be a finite set and f \colon S \rightarrow S be injective. Is f a bijection?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S be a finite set and f \colon S \rightarrow S be injective. Why must f be surjective?
Back: Otherwise f is a bijection between S and a proper subset of S, a contradiction.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S be a finite set and f \colon S \rightarrow S be surjective. Is f a bijection?
Back: Yes.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let S be a finite set and f \colon S \rightarrow S be surjective. Why must f be injective?
Back: Otherwise f is a bijection between a proper subset of S and S, a contradiction.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What does the contrapositive of the pigeonhole principle state? Back: Any set equinumerous to a proper subset of itself is infinite. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What general strategy is used to prove \omega is an infinite set?
Back: Prove \omega is equinumerous to a proper subset of itself.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Cardinal Numbers
A cardinal number is a set that is \mathop{\text{card}} A for some set A. The set \mathop{\text{card}} A is defined such that
- For any sets
AandB,\mathop{\text{card}}A = \mathop{\text{card}}BiffA \approx B. - For a finite set
A,\mathop{\text{card}}Ais the natural numbernfor whichA \approx n.
%%ANKI
Basic
How is the cardinal number of set A denoted?
Back: As \mathop{\text{card}} A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Suppose A is finite. What does \mathop{\text{card}} A evaluate to?
Back: The unique n \in \omega such that A \approx n.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Consider n \in \omega. What does \mathop{\text{card}} n evaluate to?
Back: n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose a, b, and c are distinct objects. What does \mathop{\text{card}} \{a, b, c\} evaluate to?
Back: 3
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Basic What does Enderton refer to by the "process called 'counting'"? Back: Choosing a one-to-one correspondence between two sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Cloze
A {cardinal number} is a set that is {\mathop{\text{card} } A} for some set A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
How do cardinal numbers relate to equinumerosity?
Back: For any sets A and B, \mathop{\text{card}} A = \mathop{\text{card}} B iff A \approx B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
According to Enderton, what is the "essential demand" for defining cardinal numbers?
Back: Defining cardinal numbers such that for any sets A and B, \mathop{\text{card}} A = \mathop{\text{card}} B iff A \approx B.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
What name is given to \mathop{\text{card}} \omega?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Who is attributed the assignment \mathop{\text{card}} \omega = \aleph_0?
Back: Georg Cantor.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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If one set A of cardinality \kappa is finite, then all of them are. In this case \kappa is a finite cardinal. Otherwise \kappa is an infinite cardinal.
%%ANKI
Basic
How many sets A exist such that \mathop{\text{card}} A = 0?
Back: 1
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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Basic
How many sets A exist such that \mathop{\text{card}} A = n^+ for some n \in \omega?
Back: An infinite many.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega. When is \{X \mid \mathop{\text{card}} X = n\} a set?
Back: When n = 0.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let n \in \omega. When is \{X \mid \mathop{\text{card}} X = n\} a class?
Back: Always.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
What class can we construct to prove \{X \mid \mathop{\text{card}} X = 1\} is not a set?
Back: \bigcup\, \{\{X\} \mid X \text{ is a set} \}, i.e. the union of all singleton sets.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is a finite cardinal?
Back: A cardinal number equal to \mathop{\text{card}} A for some finite set A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
What is an infinite cardinal?
Back: A cardinal number equal to \mathop{\text{card}} A for some infinite set A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
The finite cardinals are exactly what more basic set?
Back: \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
What set does \aleph_0 refer to?
Back: \mathop{\text{card}} \omega
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
What is the "smallest" infinite cardinal?
Back: \aleph_0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let C \subseteq A where A \approx n for some n \in \omega. What does \mathop{\text{card}} C evaluate to?
Back: A natural number m such that m \underline{\in} n.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Let C \subset A where A \approx n for some n \in \omega. What does \mathop{\text{card}} C evaluate to?
Back: A natural number m such that m \in n.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
How is proposition "any subset of a finite set is finite" expressed in FOL?
Back: \forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
How is the following more succinctly stated? \forall n \in \omega, \forall A \approx n, \forall B \subseteq A, \exists m \in n, B \approx m$$
Back: Any subset of a finite set is finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A and B are finite. When is A \cup B infinite?
Back: The union of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. What is the largest value \mathop{\text{card}}(A \cup B) can evaluate to?
Back: m + n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
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%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. What is the smallest value \mathop{\text{card}}(A \cup B) can evaluate to?
Back: \mathop{\text{max}}(m, n)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. When does \mathop{\text{card}}(A \cup B) = m + n?
Back: When A and B are disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. When does \mathop{\text{card}}(A \cup B) = m?
Back: When B \subseteq A.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A and B are finite. When is A \cap B finite?
Back: The intersection of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. What is the largest value \mathop{\text{card}}(A \cap B) can evaluate to?
Back: \mathop{\text{min}}(m, n)
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega s.t. A \approx m and B \approx n. What is the smallest value \mathop{\text{card}}(A \cap B) can evaluate to?
Back: 0
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Suppose sets A and B are finite. When is A \times B finite?
Back: The Cartesian product of two finite sets is always finite.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let m, n \in \omega such that A \approx m and B \approx n. What does \mathop{\text{card}}(A \times B) evaluate to?
Back: m \cdot n
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Addition
Let \kappa and \lambda be any cardinal numbers. Then \kappa + \lambda = \mathop{\text{card}}(K \cup L), where K and L are any disjoint sets of cardinality \kappa and \lambda, respectively.
%%ANKI
Basic
Let \kappa and \lambda be any cardinal numbers. How is \kappa + \lambda defined?
Back: As \mathop{\text{card}}(K \cup L) where K and L are disjoint sets with cardinality \kappa and \lambda, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. How is \mathop{\text{card}}(K \cup L) expressed in terms of cardinal numbers?
Back: N/A. K and L must be disjoint sets for this to make sense.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be disjoint sets. How is \mathop{\text{card}}(K \cup L) expressed in terms of cardinal numbers?
Back: As \kappa + \lambda where \kappa = \mathop{\text{card}} K and \lambda = \mathop{\text{card}} L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K \approx \kappa and L \approx \lambda. What is necessary for \mathop{\text{card}}(K \cup L) \approx \kappa + \lambda?
Back: That K and L are disjoint.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Addition} of cardinal numbers is defined in terms of the {union} of sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Multiplication
Let \kappa and \lambda be any cardinal numbers. Then \kappa \cdot \lambda = \mathop{\text{card}}(K \times L), where K and L are any sets of cardinality \kappa and \lambda, respectively.
%%ANKI
Basic
Let \kappa and \lambda be any cardinal numbers. How is \kappa \cdot \lambda defined?
Back: As \mathop{\text{card}}(K \times L) where K and L are sets with cardinality \kappa and \lambda, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. How is \mathop{\text{card}}(K \times L) expressed in terms of cardinal numbers?
Back: As \kappa \cdot \lambda where \kappa = \mathop{\text{card}} K and \lambda = \mathop{\text{card}} L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K \approx \kappa and L \approx \lambda. What is necessary for \mathop{\text{card}}(K \times L) \approx \kappa \cdot \lambda?
Back: N/A. This is true by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Multiplication} of cardinal numbers is defined in terms of the {Cartesian product} of sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Exponentiation
Let \kappa and \lambda be any cardinal numbers. Then \kappa^\lambda = \mathop{\text{card}}(^LK), where K and L are any sets of cardinality \kappa and \lambda, respectively.
%%ANKI
Basic
Let \kappa and \lambda be any cardinal numbers. How is \kappa^\lambda defined?
Back: As \mathop{\text{card}}(^LK) where K and L are sets with cardinality \kappa and \lambda, respectively.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. How is \mathop{\text{card}}(^LK) expressed in terms of cardinal numbers?
Back: As \kappa^\lambda where \kappa = \mathop{\text{card}} K and \lambda = \mathop{\text{card}} L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K and L be sets. How is \mathop{\text{card}}(^KL) expressed in terms of cardinal numbers?
Back: As \lambda^\kappa where \kappa = \mathop{\text{card}} K and \lambda = \mathop{\text{card}} L.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Let K \approx \kappa and L \approx \lambda. What is necessary for \mathop{\text{card}}(^LK) \approx \kappa^\lambda?
Back: N/A. This is true by definition.
Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
%%ANKI Cloze {Exponentiation} of cardinal numbers is defined in terms of the {set of functions} between sets. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
END%%
Bibliography
- Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).